Streamline methods for parabolic differential equations.

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Abstract

Parabolic advection-diffusion equations arise when modelling flow in porous media. We will in this thesis discuss two different problem set-ups from which these types of equations arise.
- Groundwater contamination with diffusion/dispersion.
- Fractional-flow formulation of immiscible two-phase flow.
Streamline methods equipped with time-off-light coordinates are attractive alternatives or supplements to traditional solution methods of advection diffusion equations. This is particulary the case when cross-streamline diffusive effects can be neglected. In this case the possibly 3-dimensional equations can be reduced to 1-dimensional equations along the streamlines . If cross-streamline effects need to be taken into account, these effects can be simulated on background grids through mappings which introduce significant numerical diffusion. We propose a method to take care of the cross-streamline diffusive effects along normallines in 2D. It is based on operator splitting, reducing the 2D equations to 1-dimensional equations along streamlines and normallines.